Bridge Scaling in Conditioned Henyey-Greenstein Random Walks

Abstract

We study fixed-length bridge paths -- half-space excursions that start and end at a planar boundary -- for three-dimensional random walks with Henyey-Greenstein scattering angles and exponentially distributed step lengths, using Monte Carlo simulation over asymmetry parameter g from 0 to 0.95 and path lengths from 4 to 200 steps. The key structural feature is that the walk evolves on a two-dimensional Markovian state space (depth, direction cosine) rather than the scalar depth coordinate alone. Four anomalies with respect to classical Brownian-excursion theory are reported. The mean amplitude scales super-diffusively, as path length to a power of 0.57--0.58 for isotropic scattering, nine standard deviations above the Brownian prediction of 0.5, with no sign of convergence out to 200 steps. The diffusion coefficient scales as the transport mean free path to the power 0.415 rather than the predicted 1.0. The midpoint depth distribution is Rayleigh rather than half-normal, consistent with a two-dimensional Bessel process. The bridge-conditioned mean direction cosine converges to minus two-thirds at the final step, independently of the asymmetry parameter and initial direction -- the classical Milne result anchored by the H-function moment identity. All anomalies are attributed to the two-dimensional state-space structure. The two anomalous exponents sum to approximately unity, suggesting a common geometric origin. Whether this constitutes a permanent universality-class shift or an anomalously slow crossover to Brownian-excursion behaviour remains the primary open question.

Figures (8)

• Figure 1: Normalised mean (left) and variance (right) profiles for $g=0.5$ and $n_s\in\{10,20,40\}$, compared with the universal parabola $4t(1-t)$ (dashed). The collapse holds for all $g$ and $n_s$ tested. The finite-$n_s$ curves lie above the parabola near $t=0$ and below it near $t=0.5$; the deviations shrink with increasing $n_s$.
• Figure 2: Peak mean depth $A(g,n_s)$ vs. $n_s$ on a log-log scale for $g\in\{0.0,0.1,\ldots,0.95\}$ (colour, light to dark). Reference slopes $n_s^{0.58}$ (dashed) and $n_s^{0.50}$ (Brownian excursion, dotted) are shown for comparison. All curves lie above the Brownian prediction for all $n_s$ studied. Slope uncertainties from the power-law fits are $\pm0.008$--$0.015$ (one standard error); the $g=0$ large-$n_s$ fit gives $\alpha_\infty=0.573\pm0.008$ (Eq. \ref{['eq:alpha_inf']}).
• Figure 3: Finite-size analysis of the local scaling exponent for $g=0$ and $g=0.1$. Left: local log-log slope $\alpha_{\mathrm{local}}(n_s)$ vs. $n_s$ on a log scale; the grey band marks $\alpha_\infty\approx0.57\pm0.02$. No convergence to $\alpha=0.5$ (dotted) is observed out to $n_s=200$. Right: the same data plotted against $1/n_s$, enabling linear extrapolation to the thermodynamic limit $1/n_s\to0$ (dashed lines, fits restricted to $n_s\ge20$). The extrapolated intercepts are $\alpha_\infty\approx0.55$ for both $g$ values, remaining $>0.5$ under this conservative estimate. The stabilisation and the finite-size extrapolation together support the conclusion that the exponent lies above the Brownian-excursion value over the entire range studied.
• Figure 4: Natural first-passage excursions. Left: First-passage length distribution $P(L)$ for $g\in \{0.0,0.5,0.9\}$, showing the expected heavy tail. Right: Mean peak depth $\langle z_{\max}\rangle$ vs. excursion length $L$ on a log-log scale for several $g$ values, with fitted exponents $\alpha$ (legend). All exponents exceed the Brownian prediction of $0.5$ (dotted line).
• Figure 5: Fitted exponent $\alpha$ as a function of bridge tolerance $\varepsilon$ for $g=0$ and $g=0.1$. The variation over a fivefold reduction in $\varepsilon$ is $\Delta\alpha<0.005$, well within statistical uncertainty. The dotted line marks the Brownian-excursion value $\alpha=0.5$. Selection bias is definitively ruled out.